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AS A-level Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Integration: further methods

Integrating standard functions, reverse chain rule, substitution, integration by parts, partial fractions, areas between curves and under parametric curves.

12 original questions, 67 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1

Find

  1. ∫ (6e3x + 5x) dx,   x > 0
  2. ∫ (3 sin 2x + 4 sec2 x) dx
  3. ∫ √(4x + 1) dx
5 marks
Show answer
a) 2e3x + 5 ln x + c b) -32 cos 2x + 4 tan x + c c) 16(4x + 1)3/2 + c
2

Find the exact value of ∫13 4xx2 + 3 dx, giving your answer in the form ln k, where k is an integer.

3 marks
Show answer
ln 9
3

Using a suitable trigonometric identity, find the exact value of

∫0π/3 4 sin2 x dx

4 marks
Show answer
2π3 - √32
4

Find ∫ x cos 3x dx

4 marks
Show answer
x3 sin 3x + 19 cos 3x + c
5

Given that k is a positive constant, show that

∫k2k x + kx2 dx

is independent of k, and state its exact value.

4 marks
Show answer
The integral equals ln 2 + 12 for every k > 0, so it is independent of k.
6

Find the exact value of

∫1e x2 ln x dx

giving your answer in the form ae3 + b, where a and b are rational.

5 marks
Show answer
29e3 + 19
7
  1. Use the substitution u2 = 2x − 1 to show that ∫15 x√(2x − 1) dx can be written as ∫13 u2 + 12 du
  2. Hence find the exact value of ∫15 x√(2x − 1) dx
6 marks
Show answer
a) shown b) 163
8

f(x) = 7x + 5(x + 2)(2x + 1),   x ≥ 0

  1. Express f(x) in partial fractions.
  2. Hence show that ∫04 f(x) dx equals ln k, stating the value of the integer k.
6 marks
Show answer
a) 3x + 2 + 12x + 1 b) ln 81, so k = 81
9

The diagram shows part of the curve with equation y = 2 sin x, 0 ≤ x ≤ π, and the line with equation y = 1

The region R, shown shaded, is bounded by the curve and the line.

  1. Find the exact x coordinates of the points where the line meets the curve.
  2. Find the exact area of R, giving your answer in the form a√3 − bπ, where a and b are rational.
xyRy = 1y = 2 sin xπO
6 marks
Show answer
a) x = π6 and x = 5π6 b) 2√3 - 2π3
10

The curve C has parametric equations

x = t2,   y = sin t,   0 ≤ t ≤ π

The region R, shown shaded in the diagram, is bounded by C and the x-axis.

  1. Show that the area of R is given by ∫0π 2t sin t dt
  2. Hence find the exact area of R.
xyRCO
7 marks
Show answer
a) shown b) 2π
11

The gates at a summer music festival open at 10 am. The rate at which people arrive, R hundred people per hour, t hours after the gates open is modelled by

R = 20t e−0.5t,   0 ≤ t ≤ 4

  1. Using differentiation, find, according to the model, the greatest rate of arrival, giving your answer in hundreds of people per hour to 3 significant figures.
  2. Show that, according to the model, the number of people who arrive in the first 4 hours is 100(80 − 240e−2)
  3. Hence find the number of people, to the nearest 10, who arrive in the first 4 hours.
8 marks
Show answer
a) 14.7 hundred people per hour (at t = 2, 12 noon) b) shown c) 4750 people
12

The diagram shows part of the curve C with equation y = 4x e−x, x ≥ 0, and the line l with equation y = x

C and l meet at the origin O and at the point A.

The finite region R, shown shaded, is bounded by C and l.

  1. Find the exact x coordinate of A.
  2. Show that the exact area of R is 3 − 2 ln 2 − 2(ln 2)2
xyRlCAO
9 marks
Show answer
a) x = ln 4 (= 2 ln 2) b) shown

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Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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